The GIT and HP methods are similar but have an essential difference. In the HP method, the solution is represented in the form of heat potential with the unknown potential density function \(\Psi (\tau )\)  which solves the corresponding Volterra equation of the second kind, see Section . In the GIT method, similar to, e.g., the Fourier method, we start with applying some integral transform to the PDE under consideration. The transform has to be such that the transformed equation with \(x \mapsto p\)  is solvable analytically in time. The second step is to construct an inverse transform, which could be computed analytically using the complex analysis. If this is possible, then the solution can be represented as an explicit integral of some kernel multiplied by the unknown function \(\Omega (\tau )\) . Hence, this looks pretty similar to the HP method. However, the function \(\Omega (\tau )\)  has now a transparent meaning - this is the gradient of the solution at the moving boundary. It turns out that this gradient also solves a Volterra equation of the second kind. As mentioned, explicit construction of such forward and inverse transforms is performed in [1]}, [2]}, [3]}, [4]}, [5]} for various models and spatial domains. Also, the authors show that the performance of both methods is the same. Both HP and GIT methods are faster than the finite-difference approach and provide higher accuracy.
